A unified account, from a pedagogical perspective, is given of the longitudinal and transverse projective delta functions proposed by Belinfante and of their relation to the Helmholtz theorem for the decomposition of a three-vector field into its lon
gitudinal and transverse components. It is argued that the results are applicable to fields that are time-dependent as well as fields that are time-independent.
A proof is given of the vector identity proposed by Gubarev, Stodolsky and Zakarov that relates the volume integral of the square of a 3-vector field to non-local integrals of the curl and divergence of the field. The identity is applied to the case
of the magnetic vector potential and magnetic field of a rotating charged shell. The latter provides a straightforward exercise in the use of the addition theorem of spherical harmonics.
The derivation of the Helmholtz theorem of vector decomposition of a 3-vector field requires that the field satisfy certain convergence properties at spatial infinity. This paper investigates if time-dependent electromagnetic radiation wave fields, w
hich are of long range, satisfy these requirements. It is found that the requirements are satisfied because the fields give rise to integrals over the radial distance r of integrands of the form sin(kr)/r and cos(kr)/r. These Dirichlet integrals converge at infinity as required.
When the electromagnetic potentials are expressed in the Coulomb gauge in terms of the electric and magnetic fields rather than the sources responsible for these fields they have a simple form that is non-local i.e. the potentials depend on the field
s at every point in space. It is this non-locality of classical electrodynamics that is at first instance responsible for the puzzle associated with the Aharonov-Bohm effect: that its interference pattern is affected by fields in a region of space that the electron beam never enters.
Foreman and Lomer proposed in 1957 a method of estimating the harmonic forces between parallel planes of atoms of primitive cubic crystals by Fourier transforming the squared frequencies of phonons propagating along principal directions. A generalize
d form of this theorem is derived in this paper and it is shown that it is more appropriate to apply the method to certain combinations of the phonon dispersion relations rather than to individual dispersion relations themselves. Further, it is also shown how the method may be extended to the non-primitive hexagonal close packed and diamond lattices. Explicit, exact and general relations in terms of atomic force constants are found for deviations from the Blackman sum rule which itself is shown to be derived from the generalized Foreman-Lomer theorem.
It is shown that when the gauge-invariant Bohr-Rosenfeld commutators of the free electromagnetic field are applied to the expressions for the linear and angular momentum of the electromagnetic field interpreted as operators then, in the absence of el
ectric and magnetic charge densities, these operators satisfy the canonical commutation relations for momentum and angular momentum. This confirms their validity as operators that can be used in quantum mechanical calculations of angular momentum.
We compare three attempts that have been made to decompose the angular momentum of the electromagnetic field into components of an orbital and spin nature. All three expressions are different and it appears, on the basis of classical electrodynamics,
that there is no preferred way of decomposing the angular momentum of the electromagnetic field into orbital and spin components, even in an inertial frame.
The instantaneous nature of the potentials of the Coulomb gauge is clarified and a concise derivation is given of the vector potential of the Coulomb gauge expressed in terms of the instantaneous magnetic field.
It is proposed that primary nucleation of amorphous microspherulites of hydrated silica in natural proto-precious-opal can be followed by a long range superlattice ordering process by means of electrostatic self-assembly. Necessary conditions in the
thermodynamics are a high surface charge density on microspherulite surfaces, a long Debye length and an appropriate number density of nucleation centres. A further chemical requirement is a high alkaline environmental pH from 9 to 10. It is also proposed that the characteristic concentric spherical shell-like structure of spherulites, centred on primary nuclei, are due to sequential deposition of intrinsic salts which precipitate out when the corresponding solubility limits in the liquid are successively exceeded. It can be that the better-known sedimentation of microspherulites under gravity only plays part in the final stabilization period of overall growth.
